Spherical varieties and Langlands duality

Let G be a connected reductive complex algebraic group. This paper is devoted to the space Z of meromorphic quasimaps from a curve into an affine spherical G-variety X. The space Z may be thought of as an algebraic model for the loop space of X. In this paper, we associate to X a connected reductive complex algebraic subgroup $\check H$ of the dual group $\check G$. The construction of $\check H$ is via Tannakian formalism: we identify a certain tensor category Q(Z) of perverse sheaves on Z with the category of finite-dimensional representations of $\check H$. Combinatorial shadows of the group $\check H$ govern many aspects of the geometry of X such as its compactifications and invariant differential operators. When X is a symmetric variety, the group $\check H$ coincides with that associated to the corresponding real form of G via the (real) geometric Satake correspondence.

设\(G\)是一个连通的约化复代数群。本文致力于研究从一条曲线到一个仿射球形\(G\)-簇\(X\)的亚纯拟映射的空间\(Z\)。空间\(Z\)可被视为\(X\)的环路空间的一个代数模型。在本文中，我们给\(X\)关联对偶群\(\check{G}\)的一个连通的约化复代数子群\(\check{H}\)。\(\check{H}\)的构造是通过坦纳凯恩形式主义：我们将\(Z\)上的某个反常层的张量范畴\(Q(Z)\)与\(\check{H}\)的有限维表示范畴等同起来。群\(\check{H}\)的组合影子支配着\(X\)的几何的许多方面，例如它的紧化和不变微分算子。当\(X\)是一个对称簇时，群\(\check{H}\)与通过（实）几何萨塔克对应关联到\(G\)的相应实形式的群一致。